AP Calculus BC 10.2 Working with Geometric Series Study Notes - New Syllabus

Working with Geometric Series

A geometric series is a series in which each term is obtained by multiplying the previous term by a constant ratio \( r \).

The general form of a geometric series is:

\( a + ar + ar^2 + ar^3 + \dots \)

where:

• \( a \) is the first term
• \( r \) is the common ratio

Sum of the First \( n \) Terms

The sum of the first \( n \) terms (partial sum) is:

\( S_n = \dfrac{a(1 – r^n)}{1 – r} \) for \( r \neq 1 \)

Sum of an Infinite Geometric Series

If \( |r| < 1 \), the series converges, and the sum is:

\( S = \dfrac{a}{1 – r} \)

If \( |r| \geq 1 \), the series diverges (sum does not exist).

Properties of Geometric Series

• If \( r > 0 \), terms keep the same sign.
• If \( r < 0 \), the terms alternate in sign.
• For \( |r| < 1 \), terms get smaller in magnitude and approach zero.